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Math and science::Analysis::Tao::06. Limits of sequences

# Square root, expressed as the limit of a sequence

Let $$c \in \mathbb{R}$$ be a real. The sequence defined recursively below converges, and it converges to $$\sqrt{c}$$.

Let $$x_1 = c$$, and let $$x_n$$ be defined like so:

$x_{n + 1} = \frac{1}{2}(x_n + \frac{c}{x_n})$

Proof by induction (any other typical methods) can be used to show that sequences defined recursively like this do (or don't) converge.

### Intuition

The outer $$\frac{1}{2}$$ factor can be seen as taking an average of the two inner terms. Inspecting the second term, $$\frac{1}{2}(c + \frac{c}{c}) = \frac{c + 1}{2}$$, the third, $$\frac{1}{2}(\frac{c+1}{2} + c \frac{2}{c+1})$$, and so on, one can be convinced that this sequences is increasing starting from the third term. In addition, it can be seen that the terms don't surpass $$\sqrt{c}$$. While not a exactly convincing as a proof, you can check if a term did surpass $$\sqrt{c}$$, then the following term would be less than that term.

Abbott, p54